536 PROCEEDINGS OF THE AMERICAN ACADEMY. 



at the point. Thus the elements of (23) are imhmitedly differentiable 

 in the neighborhood of such a point along C/c^i. The same is true at 

 more complicated points of intersection or points of contact, as a 

 similar argument shows. 



The theorem is now demonstrated, at least for the case when a 

 does not lie on a curve Ci, . . . , C,. If a does lie on such a curve, we 

 first choose an a' which is not on these curves to replace a, and then 

 resort to the simple device used at the close of § 6 to replace a' by a. 



Part II: The Problem of Riemann and its Generalization. 



§ 8. On Cauchij Matrices. 



Let r be a matrix of constants of determinant not zero, correspond- 

 ing to the coefficients of a linear transformation. According to the 

 well-known theory of classification of such matrices, based on the 

 Cayley-Weierstrass elementary divisor theory, we may write 



T = C-WC, 



where C is a matrix of constants, and in general I' is of the form (pjdij). 



Consider along with the matrix /', the matrix of functions 



I' (x) = {x^Jbi]) where 2irkj^ — 1 = logpj (j = 1,. . ., n). When x 



makes a positive circuit of x = 0, I'{x) becomes I'{x) I'. The matrix 



T ix) = /' (x) C 



will accordingly alter to 



r{x) re = I (x) c- C'U'c = t {x) t, 



when X makes such a circuit, i. e. ^(.t) will be affected by the prescribed 

 linear transformation. 



Such a matrix T{x) or a simple modification thereof exists for every 

 transformation T and is called a Cauchy matrix. ■'^'^ A certain measure 

 of arbitrariness enters into the determination of T{x) when T is given ; 

 in general k\, . . .,kn are undetermined up to an additive integer. 

 We note that the determinant of T{x) is not zero for x 5^ 0, oo . 



A final property of the Cauchy matrices which is important for us 

 is that they form the matrix solution of a linear differential system 



= LT{x), 



10 Cf. Schlesinger, Vorlesungen iiber lineare Differentialgleichungen, pp. 

 122-140, where a complete treatment is given. 



